Analysis of the Potential Field and Equilibrium Points of Irregular-shaped
Minor Celestial Bodies
Xianyu Wang1, Yu Jiang1, 2, Shengping Gong1
1. School of Aerospace Engineering, Tsinghua University, Beijing 100084, China
2. State Key Laboratory of Astronautic Dynamics, Xi’an Satellite Control Center, Xi’an 710043, China
X. Wang () e-mail: [email protected] (corresponding author)
Y. Jiang () e-mail: [email protected] (corresponding author)
Abstract. The equilibrium points of the gravitational potential field of minor celestial
bodies, including asteroids, comets, and irregular satellites of planets, are studied. In
order to understand better the orbital dynamics of massless particles moving near
celestial minor bodies and their internal structure, both internal and external
equilibrium points of the potential field of the body are analyzed. In this paper, the
location and stability of the equilibrium points of 23 minor celestial bodies are
presented. In addition, the contour plots of the gravitational effective potential of
these minor bodies are used to point out the differences between them. Furthermore,
stability and topological classifications of equilibrium points are discussed, which
clearly illustrate the topological structure near the equilibrium points and help to have
an insight into the orbital dynamics around the irregular-shaped minor celestial bodies.
The results obtained here show that there is at least one equilibrium point in the
potential field of a minor celestial body, and the number of equilibrium points could
be one, five, seven, and nine, which are all odd integers. It is found that for some
irregular-shaped celestial bodies, there are more than four equilibrium points outside
the bodies while for some others there are no external equilibrium points. If a celestial
1
body has one equilibrium point inside the body, this one is more likely linearly stable.
Key words: Minor bodies; Asteroids; Comets; Equilibrium points; Stability;
1. Introduction
During the past few decades, developments in radar shape modeling and direct
imaging have made it possible to realize the complexity of the orbital dynamics
around irregular-shaped minor celestial bodies; consistently, the topic has become the
focus of considerable interest. In this paper we focus on characterizing the dynamical
environment of irregular-shaped minor bodies by means of studying the existence,
number and stability of equilibrium points in the gravitational potential field of highly
irregular-shaped asteroids, comets and satellites of planets. A better understanding of
the stability and topological classification of the equilibrium points can help in
providing a basis to select the best reconnaissance orbit around asteroids and comets
which is essential to the success of space probes sent to study small celestial bodies.
Reliable results have been obtained by previous research devoted to investigate
the dynamical environment in the vicinity of asteroids and comets of irregular shape.
Elipe and Lara (2003) modeled the asteroid 433 Eros by a finite straight segment and
found four equilibrium points around it. Scheeres et al. (2004) used the shape and
rotating model from radar imaging data and studied the dynamical environment
associated with asteroid 25143 Itokawa; four equilibrium points were found and the
coordinates were given in the body-fixed coordinate frame. Blesa (2006) calculated
the potential field of two simple-shaped bodies, which included a triangular plate and
a square plate. Mondelo et al. (2010) found four equilibrium points around asteroid 4
2
Vesta and pointed out that there are usually four equilibrium points in the case of an
irregular gravitational potential. Liu et al. (2011) calculated the potential field of a
rotating cube and found eight equilibrium points. Furthermore, asteroids 216
Kleopatra, 1620 Geographos, 4769 Castalia and 6489 Golevka have been found to
have four equilibrium points (Yu and Baoyin 2012; Jiang et al. 2014) while asteroid
1580 Betulia has six equilibrium points and comet 67P/ Churyumov-Gerasimenko has
four equilibrium points (Scheeres 2012).
The stability of equilibrium points has a great influence on the dynamical
behavior around equilibrium points (Jiang et al. 2014). For the rotating segment, the
two collinear equilibrium points are always linearly unstable while the stability of the
two isosceles equilibrium points depends on the parameter of the rotating segment
(Elipe and Riaguas 2003). For the rotating cube, four equilibrium points are unstable
while the other four are linearly stable (Liu et al. 2011). Using precise radar data to
model asteroids, it has been established that all of the four equilibrium points outside
the asteroids 216 Kleopatra, 1620 Geographos and 4769 Castalia are unstable (Yu and
Baoyin 2012; Jiang et al. 2014); while it has been found that two equilibrium points
around asteroids 4 Vesta and 6489 Golevka are unstable, the other two being linearly
stable (Mondelo et al. 2010; Jiang et al. 2014). Previous research was mainly
concerned about the equilibrium points outside of the celestial minor body,
neglecting the inner equilibrium points. This is reasonable for planets because there is
only one internal equilibrium point and it is located almost in the center of the body.
But for the irregular-shaped celestial bodies, the situation can be more complex. The
3
nature of the equilibrium points inside a minor celestial body and their stability may
provide information on its internal structure and stresses. This information can be
used to study the failure mode of the object. Small asteroids are thought to be
gravitational aggregates of loosely consolidated material (e.g. Richardson et al. 2002).
The internal dynamical environment may help to investigate the structure of a minor
celestial body.
In this paper, we study the number of equilibrium points of the gravitational
potential of fifteen asteroids, three comets and five irregular-shaped moonlets of
planets. It is found that there exists equilibrium points inside all the celestial minor
bodies analyzed here (see Table 1). Besides, for some irregular-shaped celestial bodies,
there are more than four equilibrium points outside the celestial body whereas for
others there is no outer equilibrium point. For example, there are seven equilibrium
points in the potential field of asteroid 216 Kleopatra, four of them are outside the
body while the other three are inside it; there is only one equilibrium point in the
potential field of asteroid 1998 KY26, which is inside the body; there are nine
equilibrium points in the potential field of asteroid 101955 Bennu, eight of them are
outside the body while only one is inside it. Our results suggest that there is at least
one equilibrium point in the gravitational potential of an irregular-shaped minor body;
the majority of irregular-shaped minor bodies have five equilibrium points but there
are cases that include one, seven or nine.
Stability and topological classifications of equilibrium points are also discussed.
It is found that if the minor body has only one inner equilibrium point, such a point is
4
likely to be linearly stable. According to Jiang et al. (2014), the topological structure
of the equilibrium point can be classified into five cases. The topological
classification of the equilibrium points reveals that all of them belong to one of the
Cases 1, 2, or 5 defined in Jiang et al. (2014). The asteroids 4 Vesta, 2867 Steins, 6489
Golevka, 52760, the satellites of planets M1 Phobos, N8 Proteus, S9 Phoebe, and the
comets 1P/Halley and 9P/Tempel 1 each have three equilibrium points that belong to
Case 1. Moreover, equilibrium points which are outside the irregular-shaped minor
body, corresponding to Case 1 and 2, have a staggered distribution. On the other hand,
for other minor bodies considered here all the equilibrium points belong either to Case
2 or to Case 5 having those which are outside the minor body a staggered distribution.
2. Number and Position of Equilibrium Points
Let us consider the motion of a massless particle around a minor body. The effective
potential of the particle can be expressed as (Scheeres et al. 1996; Yu & Baoyin 2012)
V r   
where
1
 ω  r  ω  r   U  r  ,
2
(1)
is the body-fixed vector from the center of mass of the body to the particle,
ω is the rotational angular velocity vector of the body relative to the inertial frame,
and U  r  is the gravitational potential. The frame of reference used across this
paper is the body-fixed frame. The origin is in the barycenter of the minor body. The x,
y, z axes correspond to the principal axes of smallest, intermediate, and largest
moment of inertia, respectively.
Using the polyhedron method, the gravitational potential can be written as (Werner
5
and Scheeres 1996)
1
1
U  G  re gEe gre  Le  G  r f gFf gr f   f ,
2
2
eedges
f  faces
where G=6.67×10-11 m3kg-1s-2 represents the gravitational constant, σ is the density of
the polyhedron, ra (a=e, f) is a body-fixed vector from the field point to any point on
an edge (corresponding to subscript e) or face (corresponding to subscript f), Le and ωf
are factors of integration that operate over the space between the field point and edges
or faces, and Ee and Ff are dyads representing geometric parameters of edges and
faces, which are defined in terms of face- and edge-normal vectors.
The equilibrium points satisfy the following condition (Jiang et al. 2014)
V  x, y, z  V  x, y, z  V  x, y, z 


 0,
x
y
z
where
 x, y, z 
 xL , yL , zL 
V  x, y, z 
T
(2)
are the components of r in the body-fixed coordinate system. Let
denote the coordinates of the critical point; the effective potential
can be written using a Taylor expansion at the equilibrium
point  xL , yL , zL  . As for the polyhedron models of minor bodies, the potential and
T
gravity can be written as a summation form (Werner and Scheeres 1996). So Eq. (2)
can be solved numerically.
If we denote
  x  xL
  y  yL ,
  z  zL
  2V 
Vxx   2 
 x  L
  2V 
Vxy  

 xy L
  2V 
Vyy   2 
 y  L
and
  2V 
Vzz   2 
 z  L
  2V 
Vyz  
 ,
 yz  L
  2V 
Vxz  

 xz  L
6
(3)
the linearized equations of motion relative to the equilibrium point can be expressed
as (Jiang et al. 2014)
  2  Vxx  Vxy  Vxz  0
  2  Vxy  Vyy  Vyz  0 .
(4)
  Vxz  Vyz  Vzz  0
Let us consider the number of equilibrium points that satisfy the following equation
 U  r 
 2x

 x
 U  r 
 2 y .

 y
 U  r 

0
 z
(5)
The asymptotic surface of the effective potential V  V  r  is a circular cylindrical
surface when
V  
r   . This asymptotic surface is given by the equation
2 2 2
 x  y  , so the valued field of the effective potential has no lower bound.
2
On the other hand, the upper bound of the effective potential is an equilibrium point,
which suggests that there is at least one equilibrium point in the potential field of a
minor body.
If we consider minor celestial bodies which have precise radar observation data
(Thomas et al. 1996, 1997; Stooke 1997, 2002; Neese 2004; Nolan 2013), by means
of solving Eq. (2), we calculated all the equilibrium points outside and inside minor
celestial bodies presented in Table 1, finding out that the number of equilibrium points
is not a fixed value; the number depends on the actual shape of the body. Table 1
shows the number of equilibrium points in the potential field of some minor bodies.
The first fifteen rows in Table 1 correspond to asteroids, eighteenth to twentieth are
7
satellites of planets, and the last three rows correspond to comets. From Table 1, it can
be seen that there is only one equilibrium point in the potential field of asteroid 1998
KY26, which is inside the body, whereas asteroid 216 Kleopatra has seven equilibrium
points, four of them outside the body and the other three inside, and asteroid 101955
Bennu has nine equilibrium points, eight of them outside the body and the other one
inside. It is worth noting that although some minor bodies, such as 2063 Bacchus,
4769 Castalia, 25143 Itokawa, and 433 Eros, have a similar dogbone shape, only 216
Kleopatra has three inner equilibrium points. One possible reason is that only 216
Kleopatra has a really elongated neck between the two lumped masses at the ends. It
also indicates that 216 Kleopatra is less stable and it may suffer structural failure in
the future (Hirabayashi and Scheeres 2014). A deeper study of the relationship
between the inner equilibrium points of a minor body and its structural stability is
beyond the scope of the research presented here and it will be attempted in a future
paper. Regarding the remaining minor bodies presented in Table 1, each one has five
equilibrium points; four of them are outside the body while the other one is inside.
Moreover, the results in Table 1 indicate that the number of equilibrium points in the
potential field of an irregular shaped body is generally an odd number, such as one,
five, seven, or nine. Equilibrium points appear in pairs except the one that is located
near the center of the minor body.
Previous work on this subject was usually focused on the equilibrium points that
are located outside the minor body and paid no attention to the equilibrium points
inside. As for the planets or regular-shaped celestial bodies, it seems that there is only
8
one equilibrium point inside, usually located at the center of the body and therefore of
little dynamical relevance. However, in the case of irregular-shaped celestial minor
bodies, such as asteroids or comets, the inner equilibrium points may be really
meaningful given the fact that most of these minor bodies have not yet been explored
and their properties, such as surface composition and density, are based on indirect
measurement. The sources of error affecting the computation of the density are
coming from the indirect methods used to compute the masses of these minor bodies
such as the use of short-term gravitational perturbations derived from the analysis of
data on single close encounters between asteroids fitting trajectories computed for a
variety of assumed masses of the minor body under consideration to the observed path
of the other minor body, the use of spectroscopic analysis and radar albedo, the use of
long-term gravitational perturbations in the case of masses derived from periodic
variations in the relative positions of moonlets locked in stable orbital resonances, the
use of spacecraft tracking data for moonlets, asteroids and comets visited by orbiter
and flyby missions, and the use of crude computations of the masses of some comets
that have been made by estimating nongravitational forces and comparing them with
observed orbital changes. For example, the bulk density of 216 Kleopatra is 3.6±0.4
g·cm-3 and this value has been obtained from spectroscopic analysis and radar albedo
measurements that are used to calculate the density; these computations have their
own sources of error, which eventually lead to the error in the bulk density (Descamps
et al. 2011). The bulk density error of asteroids is also a result of the estimated errors
in the micro-porosity of meteorite or rubble-pile analogues (Marchis et al. 2005).
9
Moreover, there are several methods (Descamps et al. 2011; Marchis et al. 2005) to
determine the surface composition of irregular-shaped minor bodies, including such as
the use of spectral reflectance data, the use of thermal infrared spectra and thermal
radio data, the use of radar reflectivity (using observations either carried out from
Earth or from nearby spacecraft), the use of X-ray and gamma-ray fluorescence (using
measurements conducted from an orbiter or flyby spacecraft or made by landing a
probe on the body's surface), and the use of chemical analysis of surface samples
performed on samples brought to Earth by meteorites or spacecraft, or by in situ
analysis using spacecraft. The computation errors of the volume of the minor body are
coming from the sources of error associated with the above mentioned methods.
Therefore, the error in some of the physical properties of the minor bodies may be
very large.
The equilibrium points inside the body can help researchers to study the shape
evolution of these irregular-shaped minor bodies. Moreover, it has been suggested that
small asteroids are gravitational aggregates of loosely consolidated material
(Richardson et al. 2002) meaning that their internal structure is uncertain and their
inner stress and internal cohesive forces are unknown. The results obtained in this
paper can help to investigate the internal structure of an irregular-shaped minor body
by means of the knowledge of the position and stability of the internal
equilibrium points as a basis on which further research may lead to know the shape
and mass distribution of these minor bodies.
Table 1. Number of equilibrium points in the potential field of irregular-shaped
celestial minor bodies
10
Serial number
Minor bodies
Total number of equilibrium points
Outside
Inside
1
4 Vesta
5
4
1
2
216 Kleopatra
7
4
3
3
243 Ida
5
4
1
4
433 Eros
5
4
1
5
951 Gaspra
5
4
1
6
1620 Geographos
5
4
1
7
1996 HW1
5
4
1
8
1998 KY26
1
0
1
9
2063 Bacchus
5
4
1
10
2867 Steins
5
4
1
11
4769 Castalia
5
4
1
12
6489 Golevka
5
4
1
13
25143 Itokawa
5
4
1
14
52760
5
4
1
15
101955 Bennu
9
8
1
16
J5 Amalthea
5
4
1
17
M1 Phobos
5
4
1
18
N8 Proteus
5
4
1
19
S9 Phoebe
5
4
1
20
S16 Prometheus
5
4
1
21
1P/Halley
5
4
1
22
9P/Tempel1
5
4
1
23
103P/Hartley2
5
4
1
The positions of equilibrium points for them are presented in Table A1 of
Appendix 1 and the physical characteristics of these irregular-shape celestial bodies
are presented in Table A2 of Appendix 1. The data in Table A1 are determined by the
shape, rotation period and density of minor bodies. Most of the minor bodies have not
been visited by spacecraft, so the physical characters are not precise values. As for
asteroid 951 Gaspra, 1% of density error and 1% of angular velocity can cost 0.4%
and 0.8% of position error of equilibrium points, correspondingly (as show in Figure
A1 of Appendix 1). It is worth nothing that asteroid 1998 KY26 has only one
equilibrium point that is located at the center of the body. This is due to the very short
sidereal rotation period of asteroid 1998 KY26, which is only 10.704 minutes (Ostro et
al. 1999). This makes the centrifugal acceleration around it very large in the
body-fixed frame compared with the gravitational acceleration. So there is no
equilibrium outside the body. Fig. 1 shows the contour plots of the gravitational
11
effective potential and equilibrium points for minor bodies listed in Table 1. The units
of the effective potential per unit mass are m2·s-2.
a. 4 Vesta
b. 216 Kleopatra
c. 243 Ida
d. 433 Eros
e. 951 Gaspra
f. 1620 Geographos
g. 1996 HW1
h. 1998 KY26
i. 2063 Bacchus
j. 2876 Steins
k. 4769 Castalia
l. 6489 Golevka
12
m. 25143 Itokawa
n. 52760
o. 101955 Bennu
p. J5 Amalthea
q. M1 Phobos
r. N8 Proteus
s. S9 Phoebe
t. S16 Prometheus
u. 1P/Halley
v. 9P/Tempel 1
w. 103P/Hartley 2
Fig. 1. Contour plots of the gravitational effective potential per unit mass and location of
equilibrium points for minor bodies. The colour code represents the effective potential per unit
mass and the units are m2·s-2.
13
3. Stability and Topological Classifications of Equilibrium Points
Let A be the coefficient matrix of the linearized system,
 0
A 2
  V
1 
2
(6)
 Vxx Vxy Vxz 
 0  0 




2
where    0 0  , and  V   Vxy Vyy Vyz  is the Hessian matrix of the
V

 0 0 0 
 xz Vyz Vzz 
effective potential. Then Eq. (4) can be written in the following form
  A
where ε is the state variable,     
Let F  r  
(7)
T
    .
V  r 
, if the matrix  2V is positive definite, the equilibrium point
r
around the celestial body is stable (Jiang et al. 2014).
Linear classification: If the Hessian matrix 2V  τ  
F  τ 
has full rank, the
τ
equilibrium point is non-degenerate; if the rank of the Hessian matrix 2V  τ  is
less than 3, the equilibrium point is degenerate.
The Hessian matrix 2V  τ  has full rank if and only if det  2V  τ    0 , in
other words, if and only if the Hessian matrix 2V  τ  has non-zero eigenvalues.
The rank of the Hessian matrix 2V  τ  is less than three if and only if
2
det  2V  τ    0 , in other words, if and only if the Hessian matrix  V  τ  has zero
eigenvalues. The equilibrium points in the potential field of a rotating asteroid can be
classified as non-degenerate or degenerate linear.
Rank classification: Using the rank of the Hessian matrix 2V  τ  , the equilibrium
points can be classified into n  3 classes, where n is the number of columns of the
14
Hessian matrix. The rank of the Hessian matrix may be 1, 2 or 3, and each value of
the rank defines a class of the equilibrium points. Therefore, the equilibrium points in
the potential field of a rotating asteroid have three rank classes.
Different classification leads to different laws of motion when we consider the
dynamical evolution of a massless particle in the neighborhood of the equilibrium
points. Jiang et al. (2014) presented a theorem establishing eight cases of equilibrium
points in the potential field of a rotating asteroid that leads to the topological manifold
classification of the equilibrium points.
Topological manifold classification: Furthermore, for the non-degenerate and
non-resonant equilibrium points in the potential field of a celestial body, the
topological manifold classification (Jiang et al. 2014) of equilibrium points is
presented in Table 2.
Table 2. The topological manifold classification of non-degenerate and non-resonant equilibrium
points. C0: Case; C1: Eigenvalues (the imaginary eigenvalues are different); C2: Stability; C3:
Number of periodic orbit families around equilibrium points.
C0
C1
C3
LS
3
Case 1
i j   j  R  ; j  1,2,3
Case 2
 j  j  R  ; j  1  , i j   j  R  ; j  1,2

U
2
Case 3
 j  j  R  ; j  1,2  , i j   j  R  ; j  1

U
1
Case 4a
 j  j  R  ; j  1  ,   i  ,  R 
U
0
Case 4b
 j  j  R  ; j  1,2,3
U
0
U
1
Case 5
  i  ,  R 
 , i  
j
15
j

C2


 R ; j  1

Each of the irregular-shaped celestial bodies listed in Table 1 is analyzed using Eq. (6)
and Eq. (7). All of them are non-degenerate and non-resonant equilibrium points.
Therefore, the topological manifold classification is used to analyze the stability of the
equilibrium points and the results obtained are presented in Table A3 of Appendix 1.
Using the topological manifold classification, we can see from Table A3 that the
majority of the equilibrium points located at the center belong to Case 1 except that of
the asteroid 216 Kleopatra. This means that the equilibrium point near the center of
the body is usually linearly stable, which indicates a stable shape of a celestial body.
However, for asteroid 216 Kleopatra, the equilibrium point located at the center of the
body is unstable while the other two inner equilibrium points are both linearly stable.
This means that its relative tensile strength (RTS) which is close to zero, and typical
of weak and porous shattered rubble piles formed by gravitational aggregation,
prevents the body from achieving of hydrostatic equilibrium and its dogbone
shape suggests a collisional origin where two separate bodies fused together via a
gentle collision either from a low-velocity infall of fragments after a disruption event
or from tidal decay of a binary system (Magri et al. 2011).
From Tables A1 and A3 in Appendix 1, it can be seen that for each of the minor
bodies except the asteroid 216 Kleopatra, the Hessian matrix of the effective potential
at the equilibrium point near the barycenter of the body is positive definite. For most
of the minor celestial bodies, only the equilibrium point near the barycenter of the
body is linearly stable. Nevertheless, for each one of the asteroids 4 Vesta, 2867
Steins, 6489 Golevka, 52760, the satellites of planets M1 Phobos, N8 Proteus, S9
16
Phoebe, and the comets 1P/Halley and 9P/Tempel 1, there are three linearly stable
equilibrium points.
All of the equilibrium points belong to one of the Cases 1, 2, or 5. The
equilibrium points in the potential of the asteroids 243 Ida, 433 Eros, 951 Gaspra,
1620 Geographos, 1996 HW1, 2063 Bacchus, 4769 Castalia, 25143 Itokawa, 101955
Bennu, the satellites of planets J5 Amalthea, S16 Prometheus, and the comet
103P/Hartley 2, belong to Case 2 or 5; the number of equilibrium points
corresponding to Case 2 is equal to the number of equilibrium points corresponding to
Case 5. For these minor bodies, equilibrium points which are outside the body,
corresponding to Cases 2 and 5, have a staggered distribution whereas for the
asteroids 4 Vesta, 2867 Steins, 6489 Golevka, 52760, the satellites of planets M1
Phobos, N8 Proteus, S9 Phoebe, and the comets 1P/Halley and 9P/Tempel 1, outer
equilibrium points that correspond to Cases 1 and 2 also have a staggered distribution.
4. Conclusions
In this paper, we have studied the points of equilibrium of the gravitational potential
field of irregular-shaped minor celestial bodies such as asteroids, comets, and
satellites of planets. The analysis of the results obtained here indicates that there is at
least one equilibrium point in the potential field of an irregular-shaped minor body.
The number of equilibrium points outside the body is found to be likely either zero,
four, or eight but our analysis does not exclude other values. The majority of the
irregular-shaped minor bodies studied here have five equilibrium points in their
gravitational potential field. Though other values are not excluded, one, seven or nine
17
equilibrium points are also possible. In addition, stability and topological
classifications of equilibrium points are analyzed. If the celestial body has only one
equilibrium point inside its body, such a point is likely linearly stable. Our analysis
shows that, for most of the irregular-shaped bodies, the number of unstable
equilibrium points is greater than the number of linearly stable equilibrium points.
Considering the topological classifications of equilibrium points around
irregular-shaped minor celestial bodies, all of the equilibrium points belong to one of
the Cases 1, 2, or 5. For the asteroids 4 Vesta, 2867 Steins, 6489 Golevka, 52760, the
satellites of planets M1 Phobos, N8 Proteus, S9 Phoebe, as well as the comets
1P/Halley and 9P/Tempel 1, the outer equilibrium points, corresponding to Case 1 and
2, have a staggered distribution. For the remaining minor celestial bodies considered
here, all the equilibrium points belong to Case 1, 2 or 5, and the outer equilibrium
points, corresponding to Cases 2 and 5, have a staggered distribution being the
number of equilibrium points that corresponds to Case 2 equals to the one that
corresponds to Case 5. The inner equilibrium points are useful for studying the shape
evolution and mass distribution of the irregular-shaped minor body whereas the outer
equilibrium points can help to understand the orbital dynamics near the celestial
minor body.
Acknowledgements
We are grateful to the anonymous reviewer for an expeditious review and useful
comments. This research was supported by the National Basic Research Program of
China (973 Program, 2012CB720000), the State Key Laboratory Foundation of
Astronautic Dynamics (No. 2013ADL0202), and the National Natural Science
Foundation of China (11372150).
18
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Appendix 1
Table A1 Positions of the Equilibrium Points for irregular-shaped minor bodies
4 Vesta
Equilibrium Points
x (km)
y (km)
z (km)
E1
558.306
-30.0411
-1.63343
E2
-20.0372
555.912
-0.639814
E3
-558.428
-20.2773
-0.900566
E4
14.0415
-555.736
-0.527252
E5
-0.330992
-0.047349
0.722361
216 Kleopatra
Equilibrium Points
x (km)
y (km)
z (km)
E1
142.852
2.44129
1.18154
E2
-1.16383
100.740
-0.545312
E3
-144.684
5.18829
-0.272463
23
E4
2.22985
-102.102
0.271694
E5
63.4440
0.827465
-0.694572
E6
-59.5425
-0.969157
-0.191917
E7
6.21924
-0.198678
-0.308403
243 Ida
Equilibrium Points
x (km)
y (km)
z (km)
E1
31.3969
5.96274
0.0340299
E2
-2.16095
23.5734
0.0975084
E3
-33.3563
4.85067
-1.08844
E4
-1.41502
-25.4128
-0.378479
E5
5.43176
-1.41369
-0.144237
433 Eros
Equilibrium Points
x (km)
y (km)
z (km)
E1
19.1560
-2.65188
0. 142979
E2
-0. 484065
14.7247
-0.0631628
E3
-19.72858
-3.38644
0.132368
E4
-0.461655
-13.9664
-0.0743819
E5
0.549115
0.749273
-0.182043
951 Gaspra
Equilibrium Points
x (km)
y (km)
z (km)
E1
14.7323
-0.0379469
0.102217
E2
1.90075
13.0387
0.0112930
E3
-14.21262
-0.118726
0.0259255
E4
1.98791
-13.0444
0.0150812
E5
-0.692996
-0.00584606
-0.0511414
1620 Geographos
Equilibrium Points
x (km)
y (km)
z (km)
E1
2.67070
-0.0398694
0.0888751
E2
-0.142220
2.08092
-0.0220647
E3
-2.81851
-0.0557316
0.144376
E4
-0.125676
-2.04747
-0.0263415
E5
0.228201
0.0367998
-0.0315138
1996 HW1
Equilibrium Points
x (km)
y (km)
z (km)
E1
3.21197
0.133831
-0.00232722
E2
-0.150078
2.80789
0.000515378
E3
-3.26866
0.0841431
-0.00103271
24
E4
-0.181051
-2.82605
0.000146216
E5
0.452595
-0.0291869
0.00302067
1998 KY26
Equilibrium Points
x (km)
y (km)
z (km)
E1
0.00000007094
-0.00000032002
-0.0000109550
2063 Bacchus
Equilibrium Points
x (km)
y (km)
z (km)
E1
1.14738
0.0227972
-0.000861348
E2
0.0314276
1.07239
0.000711379
E3
-1.14129
0.00806235
-0.00141486
E4
0.0203102
-1.07409
0.000849894
E5
-0.0362491
-0.00393237
0.00222295
2867 Steins
Equilibrium Points
x (km)
y (km)
z (km)
E1
6.02496
-0.778495
0.0194957
E2
-0.350453
5.93496
-0.0389495
E3
-6.09443
-0.556101
0.0108389
E4
-0.258662
-5.91204
-0.0328477
E5
0.0465217
0.0118791
0.00862801
4769 Castalia
Equilibrium Points
x (km)
y (km)
z (km)
E1
0.910109
0.0228648
0.0345927
E2
-0.0427816
0.736033
0.00312877
E3
-0.953021
0.128707
0.0300658
E4
-0.0399531
-0.744131
0.00876237
E5
0.157955
-0.00144811
-0.0129416
6489 Golevka
Equilibrium Points
x (km)
y (km)
z (km)
E1
0.564128
-0.023416
-0.002882
E2
-0.571527
0.035808
-0.006081
E3
-0.021647
0.537470
-0.001060
E4
-0.026365
-0.546646
-0.000182
E5
0.002330
-0.003329
0.002198
25143 Itokawa
Equilibrium Points
x (km)
y (km)
z (km)
E1
0.554478
-0.00433107
-0.000061
25
E2
-0.0120059
0.523829
-0.000201
E3
-0.555624
-0.0103141
-0.000274
E4
-0.0158721
-0.523204
0.000246
E5
0.00346405
0.00106939
0.000105
52760
Equilibrium Points
x (km)
y (km)
z (km)
E1
1.71569
0.286648
-0.00123816
E2
-0.000226269
1.73333
-0.000340622
E3
-1.69114
0.407657
-0.00137500
E4
0.0777576
-1.73261
0.00003270
E5
0.000487075
-0.00200895
-0.000137221
101955 Bennu
Equilibrium Points
x (km)
y (km)
z (km)
E1
0.302254
0.0207971
-0.00325871
E2
0.119921
0.263907
-0.00250975
E3
-0.133380
0.265869
-0.0108739
E4
-0.211138
0.204168
-0.00979061
E5
-0.288326
-0.884228
-0.00262675
E6
-0.00230678
-0.290888
0.00159251
E7
0.133622
-0.265726
-0.00161864
E8
0.217825
-0.196538
-0.00365738
E9
0.000149629
0.00020317
0.00004969
J5 Amalthea
Equilibrium Points
x (km)
y (km)
z (km)
E1
193.626
6.99477
-0.526508
E2
-3.70510
177.930
0.503297
E3
-196.393
-9.22749
-1.68477
E4
-13.9926
-177.033
0.827095
E5
2.67129
0.497929
1.20783
M1 Phobos
Equilibrium Points
x (km)
y (km)
z (km)
E1
24.0314
-4.77124
-0.163089
E2
0.0317206
240.5832
0.703125
E3
-23.9533
-4.99313
-0.0684951
E4
0.103851
-23.9175
0.0955774
E5
-0.0767803
0.0677243
0.0811266
N8 Proteus
26
Equilibrium Points
x (km)
y (km)
z (km)
E1
887.221
-66.3922
0.309053
E2
-71.745.2
884.930
-0.00239582
E3
-890.219
-12.8162
0.402503
E4
-40.8827
-886648
-0.0538940
E5
0.400639
-0.129694
-0.522757
S9 Phoebe
Equilibrium Points
x (km)
y (km)
z (km)
E1
251.227
9.72748
0.193940
E2
-75.5341
239.050
-0.492151
E3
-250.944
21.3226
0.470713
E4
-30.0715
-248.639
-0.361233
E5
0.148101
-0.191564
-0.0134649
S16 Prometheus
Equilibrium Points
x (km)
y (km)
z (km)
E1
103.638
-7.63664
-0.206733
E2
-2.44497
94.0209
0.194146
E3
-103.999
0.106813
-1.04257
E4
1.53394
-93.7212
-0.120004
E5
-0.238211
0.351697
0.337193
1P/Halley
Equilibrium Points
x (km)
y (km)
z (km)
E1
24.9463
-0.662821
0.00372359
E2
0.944749
24.3406
-0.00137676
E3
-24.8561
-1.05235
0.00508783
E4
0.674504
-24.3224
-0.00007979
E5
-0.576790
0.142134
0.00830363
9P/Tempel 1
Equilibrium Points
x (km)
y (km)
z (km)
E1
12.9584
0.609590
-0.00574687
E2
-1.37819
12.7883
-0.00398703
E3
-12.9853
0.0332369
-0.00416586
E4
-1.71181
-12.7459
-0.00774402
E5
0.0080638
0.0054987
0.0216501
103P/Hartley 2
Equilibrium Points
x (km)
y (km)
z (km)
E1
1.48975
-0.0343398
0.00953198
27
E2
-0.142516
1.17411
-0.00293228
E3
-1.58280
-0.00593462
-0.00326151
E4
-0.137522
-1.17362
-0.00278246
E5
0.297320
0.00107410
-0.00593623
Figure A1 The x component of E1 of 951 Gaspra variation with density and angular velocity
Table A2 Physical properties of irregular-shaped celestial minor bodies
Serial number
1
Minor bodies
3
7
8
9
10
11
12
13
14
15
16
17
18
19
5.385
2.6
4.63
2.67
5.27
2.71
7.042
d
433 Eros
e1,e2
951 Gaspra
f
1620 Geographos
1996 HW1
2.0
5.223
g1,g2
3.56
8.757
h
1998 KY26
2.8
0.1784
i
2.0
14.9
j1,j2
2063 Bacchus
2867 Steins
1.8
6.04679
k1,k2
2.1
4.094
l1,l2
2.7
6.026
m1,m2
1.95
12.132
2.5
14.98
4769 Castalia
6489 Golevka
25143 Itokawa
n
52760
o1,o2
101955 Bennu
0.97
4.288
p1,p2
0.857
11.9564
q1,q2
1.876
7.65
r1,r2
1.3
26.9
s
1.63
9.27
0.48
14.71
J5 Amalthea
M1 Phobos
N8 Proteus
S9 Phoebe
t1,t2
S16 Prometheus
21
u1,u2
23
4.27
243 Ida
20
22
5.342
b1,b2
c1,c2
6
3.456
4 Vesta
216 Kleopatra
5
Rotation period(h)
a
2
4
Bulk density(g/cm3)
1P/Halley
9P/Tempel1
0.6
52.8
v1,v2
0.62
40.7
w
0.34
18.0
103P/Hartley2
28
a
Russell et al. 2012. b1 Carry et al. 2012. b2 Ostro et al. 2000. c1 Wilson et al. 1999. c2 Vokrouhlický
et al. 2003. d Yeomans et al. 2000. e1 Krasinsky et al. 2002. e2 Kaasalainen et al. 2001. f Ostro et al.
2002. g1 Magri et al. 2011. g2 Skiff et al. 2012. h Ostro et al. 1999. i Benner et al. 1999. j1 Jorda et al.
2012. j2 Keller et al. 2010. k1 Scheeres et al. 1996. k2 Hudson et al. 1997. l1 Chesley et al. 2003. l2
Hudson et al. 2000. m1 Abe et al. 2006. m2 Kaasalainen et al. 2003. n Ostro et al. 2001. o1 Chesley et
al. 2012. o2 Nolan et al. 2007. p1 Anderson et al. 2005. p2 Thomas et al. 1998. q1 Andert et al. 2010.
q2
Thayalan et al. 2008. r1 Karkoschka et al. 2003. r2 Thomas et al. 1991. s Porco et al. 2005. t1
Thomas et al. 2010. t2 Spitale et al. 2006. u1 Sagdeev et al. 1988. u2 Peale et al. 1989. v1 Britt et al.
2006. v2 A'Hearn et al. 2005. w Thomas et al. 2013.
Table A3 Topological manifold classifications of the Equilibrium Points around minor bodies. LS:
linearly stable; U: unstable;
P: positive definite; N: non-positive definite
4 Vesta
Equilibrium Points
Case
Stability
E1
2
U
V
N
E2
1
LS
P
E3
2
U
N
E4
1
LS
P
E5
1
LS
P
2
216 Kleopatra
Equilibrium Points
Case
Stability
E1
2
U
V
N
E2
5
U
N
E3
2
U
N
E4
5
U
N
E5
1
LS
P
E6
1
LS
P
E7
2
U
N
2
243 Ida
Equilibrium Points
Case
Stability
E1
2
U
V
N
E2
5
U
N
E3
2
U
N
E4
5
U
N
E5
1
LS
P
2
433 Eros
Equilibrium Points
Case
Stability
E1
2
U
V
N
E2
5
U
N
E3
2
U
N
E4
5
U
N
E5
1
LS
P
29
2
951 Gaspra
Equilibrium Points
Case
Stability
E1
2
U
V
N
E2
5
U
N
E3
2
U
N
E4
5
U
N
E5
1
LS
P
2
1620 Geographos
Equilibrium Points
Case
Stability
E1
2
U
V
N
E2
5
U
N
E3
2
U
N
E4
5
U
N
E5
1
LS
P
2
1996 HW1
Equilibrium Points
Case
Stability
E1
2
U
V
N
E2
5
U
N
E3
2
U
N
E4
5
U
N
E5
1
LS
P
V
P
2
1998 KY26
Equilibrium Points
Case
Stability
E1
1
LS
2
2063 Bacchus
Equilibrium Points
Case
Stability
E1
2
U
V
N
E2
5
U
N
E3
2
U
N
E4
5
U
N
E5
1
LS
P
2
2867 Steins
Equilibrium Points
Case
Stability
E1
2
U
V
N
E2
1
LS
P
E3
2
U
N
E4
1
LS
P
E5
1
LS
P
2
4769 Castalia
Equilibrium Points
Case
Stability
E1
2
U
V
N
E2
5
U
N
30
2
E3
2
U
N
E4
5
U
N
E5
1
LS
P
6489 Golevka
Equilibrium Points
Case
Stability
E1
2
U
V
N
E2
1
LS
N
E3
2
U
N
E4
1
LS
N
E5
1
LS
P
2
25143 Itokawa
Equilibrium Points
Case
Stability
E1
2
U
V
N
E2
5
U
N
E3
2
U
N
E4
5
U
N
E5
1
LS
P
2
52760
Equilibrium Points
Case
Stability
E1
2
U
V
N
E2
1
LS
P
E3
2
U
N
E4
1
LS
P
E5
1
LS
P
2
101955 Bennu
Equilibrium Points
Case
Stability
E1
2
U
V
N
E2
5
U
N
E3
2
U
N
E4
5
U
N
E5
2
U
N
E6
5
U
N
E7
2
U
N
E8
5
U
N
E9
1
LS
P
2
J5 Amalthea
Equilibrium Points
Case
Stability
E1
2
U
V
N
E2
5
U
N
E3
2
U
N
E4
5
U
N
E5
1
LS
P
31
2
M1 Phobos
Equilibrium Points
Case
Stability
E1
2
U
V
N
E2
1
LS
P
E3
2
U
N
E4
1
LS
P
E5
1
LS
P
2
N8 Proteus
Equilibrium Points
Case
Stability
E1
2
U
V
N
E2
1
LS
P
E3
2
U
N
E4
1
LS
P
E5
1
LS
P
2
S9 Phoebe
Equilibrium Points
Case
Stability
E1
2
U
V
N
E2
1
LS
P
E3
2
U
N
E4
1
LS
P
E5
1
LS
P
2
S16 Prometheus
Equilibrium Points
Case
Stability
E1
2
U
V
N
E2
5
U
N
E3
2
U
N
E4
5
U
N
E5
1
LS
P
2
1P/Halley
Equilibrium Points
Case
Stability
E1
2
U
V
N
E2
1
LS
N
E3
2
U
N
E4
1
LS
N
E5
1
LS
P
2
9P/Tempel1
Equilibrium Points
Case
Stability
E1
2
U
V
N
E2
1
LS
N
E3
2
U
N
E4
1
LS
N
E5
1
LS
P
32
2
103P/Hartley2
Equilibrium Points
Case
Stability
E1
2
U
V
N
E2
5
U
N
E3
2
U
N
E4
5
U
N
E5
1
LS
P
33
2